There are a few options that can work as anti-seizure medication and anti-migraine medication can help overcome the immediate effects...none of them are permanent solutions though. The key thing is you want to clear up the headache and nausea to allow for some cognitive thought.
One key thing to note that the brain can be considered a muscle in this sense...an injured muscle hurts to use and the best treatment is...well, not to use it. You might be able to get some short term cognitive thought, but the longer this person attempts to put together cohesive thought the worse they are going to be. Getting them to do their thinking in short bursts before returning to rest is the best choice. Optical functions, especially in regards to smartphones or computer screens, will cause pain and dizziness pretty quickly...if they need to use a computer, best to have another doing the typing and reading off the screen for them.
I should note that I really don't recommend any of this, it's somewhat theory, and long term effects are completely ignored, and probably does nothing beyond addressing some of the immediate symptoms. Rest and time is truly the best answer...Most medical sites won't list a lot of this because it opens them up to liability by giving a short term solution that causes long term issues instead.
First...avoid anything that ultimately acts as a blood thinner, including asprin and some other headache medication. Thinning the blood is very dangerous in concussion cases. Second, in most head injury treatments, drugs are horribly frown upon...increasing blood pressure or thinning blood is an absolute no no and needs to be avoided at all costs.
Amitriptyline - This is an anti-depressant that one of your inmates can be on. It resolves the nausea and dizziness and provides some relief as far as the head ache would go. When combined with Innopran, it can be used to treat migraines. There's a few other anti-depressants that fit into the same family that may be helpful here, but no guarantees
Paracetamol - More well known as the acetaminophen family...ya, I'm recommending taking 2 Tylenol. This will help with pain and it won't really make the concussion worse...not sure if it helps nausea and dizziness much. Unfortunately, the numbed pain isn't a good thing...pain tells you when you are further injuring yourself and a owner of a numbed brain can take on activities that damage itself further.
Marijuana - Not sure if the state your character is left in is all too helpful, but this does work to control the nausea and headache effects for a couple hours before they need to light up again.
Gabapentin - This is a anti-seizure medication that can also be used to treat nerve pain. It'll address the dizziness, the headache, and nausea...but it can cause extreme drowsiness which might be counter-intuitive to your situation.
Topiramate - This is an anti-epilepsy drug that will kill the headache and dizziness/nausea. Worst side effect here is weightloss ;)
Carbamazepine - similiar to above...anti-epilepsy medications seem to have the best chance have having some effect here.
dextroamphetamine - hesitate to put this one here...but this is a potent nervous system stimulant that is now prescribed to ADHD peeps. It was used back in WWII and given to soldiers to fight the effects of fatigue and nausea. It may help provide some short term cognitive functions, but this might be exceedingly harmful in the long run as it may interfere with brains ability to heal.
I'll end with a disclaimer since I really don't want to be sued. None of this is recommended and is only intended for story use. Concussions are serious, don't play around with the future of your brain unless you are planning on dying within a week.
Edit : Some illicits:
MDMA (also known as E or ecstasy) would probably have a short term gain allowing the person here to collect some thoughts and function. Admittedly they'll be high and potentially want to touch anything that looks soft, but the direct symptoms of the concussion should be addressed. It's not that the concussion isn't there, the brain is overwhelmed and just too high to figure it out how damaged it is. This isn't treatment at all and the long term effects are probably harshly negative, but short term it'll leave the person functioning. It'd take around 30 minutes after taking for the effects to kick in and probably put the person onto high / low waves (high they'll have no clue they're injured, lows will let them remember how much pain they are in) for a good 6 to 12 hours.
Cocaine. Fits into above category, but snorting blow will put a numbing agent directly in the brain. Without question, the long term effects here are exceedingly dangerous, but short term might give the person some capacity of cognitive thought. It's much more instantaneous, but I don't think the effects would last much past an hour.
Once again, in no way would I ever recommend the two options above...call these musings at best.